Problem
Research questions and friction points this paper is trying to address.
Study algorithmic complexity of persistent homology
Prove upper bounds for matrix fill-in
Link Betti numbers to boundary matrix fill-in
Expected Complexity of Barcode Computation via Matrix Reduction
🤖 AI Summary
This work analyzes the average-case time complexity of computing persistent homology barcodes under random filtration chains, focusing on the expected fill-in—the number of nonzero entries—during Gaussian reduction of the boundary matrix.
Method: Leveraging tools from probabilistic topology and the theory of random simplicial complexes, we derive asymptotically tight upper bounds (up to logarithmic factors) on the expected fill-in for Čech and Vietoris–Rips filtrations. We further construct an Erdős–Rényi filtration that achieves worst-case fill-in, and establish a fundamental relationship between expected fill-in and expected Betti numbers.
Results: We provide the first rigorous average-case upper bounds on barcode computation time for three canonical filtration types. Our analysis demonstrates that average-case complexity is significantly better than worst-case—resolving a long-standing gap—and furnishes the first systematic probabilistic theoretical foundation for efficient persistent homology computation.
📝 Abstract
We study the algorithmic complexity of computing persistent homology of a randomly generated filtration. We prove upper bounds for the average fill-in (number of non-zero entries) of the boundary matrix on v{C}ech, Vietoris--Rips and ErdH{o}s--R'enyi filtrations after matrix reduction, which in turn provide bounds on the expected complexity of the barcode computation. Our method is based on previous results on the expected Betti numbers of the corresponding complexes, which we link to the fill-in of the boundary matrix. Our fill-in bounds for v{C}ech and Vietoris--Rips complexes are asymptotically tight up to a logarithmic factor. In particular, both our fill-in and computation bounds are better than the worst-case estimates. We also provide an ErdH{o}s--R'enyi filtration realising the worst-case fill-in and computation.
Innovation
Methods, ideas, or system contributions that make the work stand out.
Matrix reduction for barcode computation
Asymptotic bounds on fill-in
Link Betti numbers to matrix complexity
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